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"""Non-negative matrix factorization."""
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
import itertools
import time
import warnings
from abc import ABC
from math import sqrt
from numbers import Integral, Real
import numpy as np
import scipy.sparse as sp
from scipy import linalg
from .._config import config_context
from ..base import (
BaseEstimator,
ClassNamePrefixFeaturesOutMixin,
TransformerMixin,
_fit_context,
)
from ..exceptions import ConvergenceWarning
from ..utils import check_array, check_random_state, gen_batches, metadata_routing
from ..utils._param_validation import (
Interval,
StrOptions,
validate_params,
)
from ..utils.deprecation import _deprecate_Xt_in_inverse_transform
from ..utils.extmath import randomized_svd, safe_sparse_dot, squared_norm
from ..utils.validation import (
check_is_fitted,
check_non_negative,
validate_data,
)
from ._cdnmf_fast import _update_cdnmf_fast
EPSILON = np.finfo(np.float32).eps
def norm(x):
"""Dot product-based Euclidean norm implementation.
See: http://fa.bianp.net/blog/2011/computing-the-vector-norm/
Parameters
----------
x : array-like
Vector for which to compute the norm.
"""
return sqrt(squared_norm(x))
def trace_dot(X, Y):
"""Trace of np.dot(X, Y.T).
Parameters
----------
X : array-like
First matrix.
Y : array-like
Second matrix.
"""
return np.dot(X.ravel(), Y.ravel())
def _check_init(A, shape, whom):
A = check_array(A)
if shape[0] != "auto" and A.shape[0] != shape[0]:
raise ValueError(
f"Array with wrong first dimension passed to {whom}. Expected {shape[0]}, "
f"but got {A.shape[0]}."
)
if shape[1] != "auto" and A.shape[1] != shape[1]:
raise ValueError(
f"Array with wrong second dimension passed to {whom}. Expected {shape[1]}, "
f"but got {A.shape[1]}."
)
check_non_negative(A, whom)
if np.max(A) == 0:
raise ValueError(f"Array passed to {whom} is full of zeros.")
def _beta_divergence(X, W, H, beta, square_root=False):
"""Compute the beta-divergence of X and dot(W, H).
Parameters
----------
X : float or array-like of shape (n_samples, n_features)
W : float or array-like of shape (n_samples, n_components)
H : float or array-like of shape (n_components, n_features)
beta : float or {'frobenius', 'kullback-leibler', 'itakura-saito'}
Parameter of the beta-divergence.
If beta == 2, this is half the Frobenius *squared* norm.
If beta == 1, this is the generalized Kullback-Leibler divergence.
If beta == 0, this is the Itakura-Saito divergence.
Else, this is the general beta-divergence.
square_root : bool, default=False
If True, return np.sqrt(2 * res)
For beta == 2, it corresponds to the Frobenius norm.
Returns
-------
res : float
Beta divergence of X and np.dot(X, H).
"""
beta = _beta_loss_to_float(beta)
# The method can be called with scalars
if not sp.issparse(X):
X = np.atleast_2d(X)
W = np.atleast_2d(W)
H = np.atleast_2d(H)
# Frobenius norm
if beta == 2:
# Avoid the creation of the dense np.dot(W, H) if X is sparse.
if sp.issparse(X):
norm_X = np.dot(X.data, X.data)
norm_WH = trace_dot(np.linalg.multi_dot([W.T, W, H]), H)
cross_prod = trace_dot((X @ H.T), W)
res = (norm_X + norm_WH - 2.0 * cross_prod) / 2.0
else:
res = squared_norm(X - np.dot(W, H)) / 2.0
if square_root:
return np.sqrt(res * 2)
else:
return res
if sp.issparse(X):
# compute np.dot(W, H) only where X is nonzero
WH_data = _special_sparse_dot(W, H, X).data
X_data = X.data
else:
WH = np.dot(W, H)
WH_data = WH.ravel()
X_data = X.ravel()
# do not affect the zeros: here 0 ** (-1) = 0 and not infinity
indices = X_data > EPSILON
WH_data = WH_data[indices]
X_data = X_data[indices]
# used to avoid division by zero
WH_data[WH_data < EPSILON] = EPSILON
# generalized Kullback-Leibler divergence
if beta == 1:
# fast and memory efficient computation of np.sum(np.dot(W, H))
sum_WH = np.dot(np.sum(W, axis=0), np.sum(H, axis=1))
# computes np.sum(X * log(X / WH)) only where X is nonzero
div = X_data / WH_data
res = np.dot(X_data, np.log(div))
# add full np.sum(np.dot(W, H)) - np.sum(X)
res += sum_WH - X_data.sum()
# Itakura-Saito divergence
elif beta == 0:
div = X_data / WH_data
res = np.sum(div) - np.prod(X.shape) - np.sum(np.log(div))
# beta-divergence, beta not in (0, 1, 2)
else:
if sp.issparse(X):
# slow loop, but memory efficient computation of :
# np.sum(np.dot(W, H) ** beta)
sum_WH_beta = 0
for i in range(X.shape[1]):
sum_WH_beta += np.sum(np.dot(W, H[:, i]) ** beta)
else:
sum_WH_beta = np.sum(WH**beta)
sum_X_WH = np.dot(X_data, WH_data ** (beta - 1))
res = (X_data**beta).sum() - beta * sum_X_WH
res += sum_WH_beta * (beta - 1)
res /= beta * (beta - 1)
if square_root:
res = max(res, 0) # avoid negative number due to rounding errors
return np.sqrt(2 * res)
else:
return res
def _special_sparse_dot(W, H, X):
"""Computes np.dot(W, H), only where X is non zero."""
if sp.issparse(X):
ii, jj = X.nonzero()
n_vals = ii.shape[0]
dot_vals = np.empty(n_vals)
n_components = W.shape[1]
batch_size = max(n_components, n_vals // n_components)
for start in range(0, n_vals, batch_size):
batch = slice(start, start + batch_size)
dot_vals[batch] = np.multiply(W[ii[batch], :], H.T[jj[batch], :]).sum(
axis=1
)
WH = sp.coo_matrix((dot_vals, (ii, jj)), shape=X.shape)
return WH.tocsr()
else:
return np.dot(W, H)
def _beta_loss_to_float(beta_loss):
"""Convert string beta_loss to float."""
beta_loss_map = {"frobenius": 2, "kullback-leibler": 1, "itakura-saito": 0}
if isinstance(beta_loss, str):
beta_loss = beta_loss_map[beta_loss]
return beta_loss
def _initialize_nmf(X, n_components, init=None, eps=1e-6, random_state=None):
"""Algorithms for NMF initialization.
Computes an initial guess for the non-negative
rank k matrix approximation for X: X = WH.
Parameters
----------
X : array-like of shape (n_samples, n_features)
The data matrix to be decomposed.
n_components : int
The number of components desired in the approximation.
init : {'random', 'nndsvd', 'nndsvda', 'nndsvdar'}, default=None
Method used to initialize the procedure.
Valid options:
- None: 'nndsvda' if n_components <= min(n_samples, n_features),
otherwise 'random'.
- 'random': non-negative random matrices, scaled with:
sqrt(X.mean() / n_components)
- 'nndsvd': Nonnegative Double Singular Value Decomposition (NNDSVD)
initialization (better for sparseness)
- 'nndsvda': NNDSVD with zeros filled with the average of X
(better when sparsity is not desired)
- 'nndsvdar': NNDSVD with zeros filled with small random values
(generally faster, less accurate alternative to NNDSVDa
for when sparsity is not desired)
- 'custom': use custom matrices W and H
.. versionchanged:: 1.1
When `init=None` and n_components is less than n_samples and n_features
defaults to `nndsvda` instead of `nndsvd`.
eps : float, default=1e-6
Truncate all values less then this in output to zero.
random_state : int, RandomState instance or None, default=None
Used when ``init`` == 'nndsvdar' or 'random'. Pass an int for
reproducible results across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
W : array-like of shape (n_samples, n_components)
Initial guesses for solving X ~= WH.
H : array-like of shape (n_components, n_features)
Initial guesses for solving X ~= WH.
References
----------
C. Boutsidis, E. Gallopoulos: SVD based initialization: A head start for
nonnegative matrix factorization - Pattern Recognition, 2008
http://tinyurl.com/nndsvd
"""
check_non_negative(X, "NMF initialization")
n_samples, n_features = X.shape
if (
init is not None
and init != "random"
and n_components > min(n_samples, n_features)
):
raise ValueError(
"init = '{}' can only be used when "
"n_components <= min(n_samples, n_features)".format(init)
)
if init is None:
if n_components <= min(n_samples, n_features):
init = "nndsvda"
else:
init = "random"
# Random initialization
if init == "random":
avg = np.sqrt(X.mean() / n_components)
rng = check_random_state(random_state)
H = avg * rng.standard_normal(size=(n_components, n_features)).astype(
X.dtype, copy=False
)
W = avg * rng.standard_normal(size=(n_samples, n_components)).astype(
X.dtype, copy=False
)
np.abs(H, out=H)
np.abs(W, out=W)
return W, H
# NNDSVD initialization
U, S, V = randomized_svd(X, n_components, random_state=random_state)
W = np.zeros_like(U)
H = np.zeros_like(V)
# The leading singular triplet is non-negative
# so it can be used as is for initialization.
W[:, 0] = np.sqrt(S[0]) * np.abs(U[:, 0])
H[0, :] = np.sqrt(S[0]) * np.abs(V[0, :])
for j in range(1, n_components):
x, y = U[:, j], V[j, :]
# extract positive and negative parts of column vectors
x_p, y_p = np.maximum(x, 0), np.maximum(y, 0)
x_n, y_n = np.abs(np.minimum(x, 0)), np.abs(np.minimum(y, 0))
# and their norms
x_p_nrm, y_p_nrm = norm(x_p), norm(y_p)
x_n_nrm, y_n_nrm = norm(x_n), norm(y_n)
m_p, m_n = x_p_nrm * y_p_nrm, x_n_nrm * y_n_nrm
# choose update
if m_p > m_n:
u = x_p / x_p_nrm
v = y_p / y_p_nrm
sigma = m_p
else:
u = x_n / x_n_nrm
v = y_n / y_n_nrm
sigma = m_n
lbd = np.sqrt(S[j] * sigma)
W[:, j] = lbd * u
H[j, :] = lbd * v
W[W < eps] = 0
H[H < eps] = 0
if init == "nndsvd":
pass
elif init == "nndsvda":
avg = X.mean()
W[W == 0] = avg
H[H == 0] = avg
elif init == "nndsvdar":
rng = check_random_state(random_state)
avg = X.mean()
W[W == 0] = abs(avg * rng.standard_normal(size=len(W[W == 0])) / 100)
H[H == 0] = abs(avg * rng.standard_normal(size=len(H[H == 0])) / 100)
else:
raise ValueError(
"Invalid init parameter: got %r instead of one of %r"
% (init, (None, "random", "nndsvd", "nndsvda", "nndsvdar"))
)
return W, H
def _update_coordinate_descent(X, W, Ht, l1_reg, l2_reg, shuffle, random_state):
"""Helper function for _fit_coordinate_descent.
Update W to minimize the objective function, iterating once over all
coordinates. By symmetry, to update H, one can call
_update_coordinate_descent(X.T, Ht, W, ...).
"""
n_components = Ht.shape[1]
HHt = np.dot(Ht.T, Ht)
XHt = safe_sparse_dot(X, Ht)
# L2 regularization corresponds to increase of the diagonal of HHt
if l2_reg != 0.0:
# adds l2_reg only on the diagonal
HHt.flat[:: n_components + 1] += l2_reg
# L1 regularization corresponds to decrease of each element of XHt
if l1_reg != 0.0:
XHt -= l1_reg
if shuffle:
permutation = random_state.permutation(n_components)
else:
permutation = np.arange(n_components)
# The following seems to be required on 64-bit Windows w/ Python 3.5.
permutation = np.asarray(permutation, dtype=np.intp)
return _update_cdnmf_fast(W, HHt, XHt, permutation)
def _fit_coordinate_descent(
X,
W,
H,
tol=1e-4,
max_iter=200,
l1_reg_W=0,
l1_reg_H=0,
l2_reg_W=0,
l2_reg_H=0,
update_H=True,
verbose=0,
shuffle=False,
random_state=None,
):
"""Compute Non-negative Matrix Factorization (NMF) with Coordinate Descent
The objective function is minimized with an alternating minimization of W
and H. Each minimization is done with a cyclic (up to a permutation of the
features) Coordinate Descent.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Constant matrix.
W : array-like of shape (n_samples, n_components)
Initial guess for the solution.
H : array-like of shape (n_components, n_features)
Initial guess for the solution.
tol : float, default=1e-4
Tolerance of the stopping condition.
max_iter : int, default=200
Maximum number of iterations before timing out.
l1_reg_W : float, default=0.
L1 regularization parameter for W.
l1_reg_H : float, default=0.
L1 regularization parameter for H.
l2_reg_W : float, default=0.
L2 regularization parameter for W.
l2_reg_H : float, default=0.
L2 regularization parameter for H.
update_H : bool, default=True
Set to True, both W and H will be estimated from initial guesses.
Set to False, only W will be estimated.
verbose : int, default=0
The verbosity level.
shuffle : bool, default=False
If true, randomize the order of coordinates in the CD solver.
random_state : int, RandomState instance or None, default=None
Used to randomize the coordinates in the CD solver, when
``shuffle`` is set to ``True``. Pass an int for reproducible
results across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
W : ndarray of shape (n_samples, n_components)
Solution to the non-negative least squares problem.
H : ndarray of shape (n_components, n_features)
Solution to the non-negative least squares problem.
n_iter : int
The number of iterations done by the algorithm.
References
----------
.. [1] :doi:`"Fast local algorithms for large scale nonnegative matrix and tensor
factorizations" <10.1587/transfun.E92.A.708>`
Cichocki, Andrzej, and P. H. A. N. Anh-Huy. IEICE transactions on fundamentals
of electronics, communications and computer sciences 92.3: 708-721, 2009.
"""
# so W and Ht are both in C order in memory
Ht = check_array(H.T, order="C")
X = check_array(X, accept_sparse="csr")
rng = check_random_state(random_state)
for n_iter in range(1, max_iter + 1):
violation = 0.0
# Update W
violation += _update_coordinate_descent(
X, W, Ht, l1_reg_W, l2_reg_W, shuffle, rng
)
# Update H
if update_H:
violation += _update_coordinate_descent(
X.T, Ht, W, l1_reg_H, l2_reg_H, shuffle, rng
)
if n_iter == 1:
violation_init = violation
if violation_init == 0:
break
if verbose:
print("violation:", violation / violation_init)
if violation / violation_init <= tol:
if verbose:
print("Converged at iteration", n_iter + 1)
break
return W, Ht.T, n_iter
def _multiplicative_update_w(
X,
W,
H,
beta_loss,
l1_reg_W,
l2_reg_W,
gamma,
H_sum=None,
HHt=None,
XHt=None,
update_H=True,
):
"""Update W in Multiplicative Update NMF."""
if beta_loss == 2:
# Numerator
if XHt is None:
XHt = safe_sparse_dot(X, H.T)
if update_H:
# avoid a copy of XHt, which will be re-computed (update_H=True)
numerator = XHt
else:
# preserve the XHt, which is not re-computed (update_H=False)
numerator = XHt.copy()
# Denominator
if HHt is None:
HHt = np.dot(H, H.T)
denominator = np.dot(W, HHt)
else:
# Numerator
# if X is sparse, compute WH only where X is non zero
WH_safe_X = _special_sparse_dot(W, H, X)
if sp.issparse(X):
WH_safe_X_data = WH_safe_X.data
X_data = X.data
else:
WH_safe_X_data = WH_safe_X
X_data = X
# copy used in the Denominator
WH = WH_safe_X.copy()
if beta_loss - 1.0 < 0:
WH[WH < EPSILON] = EPSILON
# to avoid taking a negative power of zero
if beta_loss - 2.0 < 0:
WH_safe_X_data[WH_safe_X_data < EPSILON] = EPSILON
if beta_loss == 1:
np.divide(X_data, WH_safe_X_data, out=WH_safe_X_data)
elif beta_loss == 0:
# speeds up computation time
# refer to /numpy/numpy/issues/9363
WH_safe_X_data **= -1
WH_safe_X_data **= 2
# element-wise multiplication
WH_safe_X_data *= X_data
else:
WH_safe_X_data **= beta_loss - 2
# element-wise multiplication
WH_safe_X_data *= X_data
# here numerator = dot(X * (dot(W, H) ** (beta_loss - 2)), H.T)
numerator = safe_sparse_dot(WH_safe_X, H.T)
# Denominator
if beta_loss == 1:
if H_sum is None:
H_sum = np.sum(H, axis=1) # shape(n_components, )
denominator = H_sum[np.newaxis, :]
else:
# computation of WHHt = dot(dot(W, H) ** beta_loss - 1, H.T)
if sp.issparse(X):
# memory efficient computation
# (compute row by row, avoiding the dense matrix WH)
WHHt = np.empty(W.shape)
for i in range(X.shape[0]):
WHi = np.dot(W[i, :], H)
if beta_loss - 1 < 0:
WHi[WHi < EPSILON] = EPSILON
WHi **= beta_loss - 1
WHHt[i, :] = np.dot(WHi, H.T)
else:
WH **= beta_loss - 1
WHHt = np.dot(WH, H.T)
denominator = WHHt
# Add L1 and L2 regularization
if l1_reg_W > 0:
denominator += l1_reg_W
if l2_reg_W > 0:
denominator = denominator + l2_reg_W * W
denominator[denominator == 0] = EPSILON
numerator /= denominator
delta_W = numerator
# gamma is in ]0, 1]
if gamma != 1:
delta_W **= gamma
W *= delta_W
return W, H_sum, HHt, XHt
def _multiplicative_update_h(
X, W, H, beta_loss, l1_reg_H, l2_reg_H, gamma, A=None, B=None, rho=None
):
"""update H in Multiplicative Update NMF."""
if beta_loss == 2:
numerator = safe_sparse_dot(W.T, X)
denominator = np.linalg.multi_dot([W.T, W, H])
else:
# Numerator
WH_safe_X = _special_sparse_dot(W, H, X)
if sp.issparse(X):
WH_safe_X_data = WH_safe_X.data
X_data = X.data
else:
WH_safe_X_data = WH_safe_X
X_data = X
# copy used in the Denominator
WH = WH_safe_X.copy()
if beta_loss - 1.0 < 0:
WH[WH < EPSILON] = EPSILON
# to avoid division by zero
if beta_loss - 2.0 < 0:
WH_safe_X_data[WH_safe_X_data < EPSILON] = EPSILON
if beta_loss == 1:
np.divide(X_data, WH_safe_X_data, out=WH_safe_X_data)
elif beta_loss == 0:
# speeds up computation time
# refer to /numpy/numpy/issues/9363
WH_safe_X_data **= -1
WH_safe_X_data **= 2
# element-wise multiplication
WH_safe_X_data *= X_data
else:
WH_safe_X_data **= beta_loss - 2
# element-wise multiplication
WH_safe_X_data *= X_data
# here numerator = dot(W.T, (dot(W, H) ** (beta_loss - 2)) * X)
numerator = safe_sparse_dot(W.T, WH_safe_X)
# Denominator
if beta_loss == 1:
W_sum = np.sum(W, axis=0) # shape(n_components, )
W_sum[W_sum == 0] = 1.0
denominator = W_sum[:, np.newaxis]
# beta_loss not in (1, 2)
else:
# computation of WtWH = dot(W.T, dot(W, H) ** beta_loss - 1)
if sp.issparse(X):
# memory efficient computation
# (compute column by column, avoiding the dense matrix WH)
WtWH = np.empty(H.shape)
for i in range(X.shape[1]):
WHi = np.dot(W, H[:, i])
if beta_loss - 1 < 0:
WHi[WHi < EPSILON] = EPSILON
WHi **= beta_loss - 1
WtWH[:, i] = np.dot(W.T, WHi)
else:
WH **= beta_loss - 1
WtWH = np.dot(W.T, WH)
denominator = WtWH
# Add L1 and L2 regularization
if l1_reg_H > 0:
denominator += l1_reg_H
if l2_reg_H > 0:
denominator = denominator + l2_reg_H * H
denominator[denominator == 0] = EPSILON
if A is not None and B is not None:
# Updates for the online nmf
if gamma != 1:
H **= 1 / gamma
numerator *= H
A *= rho
B *= rho
A += numerator
B += denominator
H = A / B
if gamma != 1:
H **= gamma
else:
delta_H = numerator
delta_H /= denominator
if gamma != 1:
delta_H **= gamma
H *= delta_H
return H
def _fit_multiplicative_update(
X,
W,
H,
beta_loss="frobenius",
max_iter=200,
tol=1e-4,
l1_reg_W=0,
l1_reg_H=0,
l2_reg_W=0,
l2_reg_H=0,
update_H=True,
verbose=0,
):
"""Compute Non-negative Matrix Factorization with Multiplicative Update.
The objective function is _beta_divergence(X, WH) and is minimized with an
alternating minimization of W and H. Each minimization is done with a
Multiplicative Update.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Constant input matrix.
W : array-like of shape (n_samples, n_components)
Initial guess for the solution.
H : array-like of shape (n_components, n_features)
Initial guess for the solution.
beta_loss : float or {'frobenius', 'kullback-leibler', \
'itakura-saito'}, default='frobenius'
String must be in {'frobenius', 'kullback-leibler', 'itakura-saito'}.
Beta divergence to be minimized, measuring the distance between X
and the dot product WH. Note that values different from 'frobenius'
(or 2) and 'kullback-leibler' (or 1) lead to significantly slower
fits. Note that for beta_loss <= 0 (or 'itakura-saito'), the input
matrix X cannot contain zeros.
max_iter : int, default=200
Number of iterations.
tol : float, default=1e-4
Tolerance of the stopping condition.
l1_reg_W : float, default=0.
L1 regularization parameter for W.
l1_reg_H : float, default=0.
L1 regularization parameter for H.
l2_reg_W : float, default=0.
L2 regularization parameter for W.
l2_reg_H : float, default=0.
L2 regularization parameter for H.
update_H : bool, default=True
Set to True, both W and H will be estimated from initial guesses.
Set to False, only W will be estimated.
verbose : int, default=0
The verbosity level.
Returns
-------
W : ndarray of shape (n_samples, n_components)
Solution to the non-negative least squares problem.
H : ndarray of shape (n_components, n_features)
Solution to the non-negative least squares problem.
n_iter : int
The number of iterations done by the algorithm.
References
----------
Lee, D. D., & Seung, H., S. (2001). Algorithms for Non-negative Matrix
Factorization. Adv. Neural Inform. Process. Syst.. 13.
Fevotte, C., & Idier, J. (2011). Algorithms for nonnegative matrix
factorization with the beta-divergence. Neural Computation, 23(9).
"""
start_time = time.time()
beta_loss = _beta_loss_to_float(beta_loss)
# gamma for Maximization-Minimization (MM) algorithm [Fevotte 2011]
if beta_loss < 1:
gamma = 1.0 / (2.0 - beta_loss)
elif beta_loss > 2:
gamma = 1.0 / (beta_loss - 1.0)
else:
gamma = 1.0
# used for the convergence criterion
error_at_init = _beta_divergence(X, W, H, beta_loss, square_root=True)
previous_error = error_at_init
H_sum, HHt, XHt = None, None, None
for n_iter in range(1, max_iter + 1):
# update W
# H_sum, HHt and XHt are saved and reused if not update_H
W, H_sum, HHt, XHt = _multiplicative_update_w(
X,
W,
H,
beta_loss=beta_loss,
l1_reg_W=l1_reg_W,
l2_reg_W=l2_reg_W,
gamma=gamma,
H_sum=H_sum,
HHt=HHt,
XHt=XHt,
update_H=update_H,
)
# necessary for stability with beta_loss < 1
if beta_loss < 1:
W[W < np.finfo(np.float64).eps] = 0.0
# update H (only at fit or fit_transform)
if update_H:
H = _multiplicative_update_h(
X,
W,
H,
beta_loss=beta_loss,
l1_reg_H=l1_reg_H,
l2_reg_H=l2_reg_H,
gamma=gamma,
)
# These values will be recomputed since H changed
H_sum, HHt, XHt = None, None, None
# necessary for stability with beta_loss < 1
if beta_loss <= 1:
H[H < np.finfo(np.float64).eps] = 0.0
# test convergence criterion every 10 iterations
if tol > 0 and n_iter % 10 == 0:
error = _beta_divergence(X, W, H, beta_loss, square_root=True)
if verbose:
iter_time = time.time()
print(
"Epoch %02d reached after %.3f seconds, error: %f"
% (n_iter, iter_time - start_time, error)
)
if (previous_error - error) / error_at_init < tol:
break
previous_error = error
# do not print if we have already printed in the convergence test
if verbose and (tol == 0 or n_iter % 10 != 0):
end_time = time.time()
print(
"Epoch %02d reached after %.3f seconds." % (n_iter, end_time - start_time)
)
return W, H, n_iter
@validate_params(
{
"X": ["array-like", "sparse matrix"],
"W": ["array-like", None],
"H": ["array-like", None],
"update_H": ["boolean"],
},
prefer_skip_nested_validation=False,
)
def non_negative_factorization(
X,
W=None,
H=None,
n_components="auto",
*,
init=None,
update_H=True,
solver="cd",
beta_loss="frobenius",
tol=1e-4,
max_iter=200,
alpha_W=0.0,
alpha_H="same",
l1_ratio=0.0,
random_state=None,
verbose=0,
shuffle=False,
):
"""Compute Non-negative Matrix Factorization (NMF).
Find two non-negative matrices (W, H) whose product approximates the non-
negative matrix X. This factorization can be used for example for
dimensionality reduction, source separation or topic extraction.
The objective function is:
.. math::
L(W, H) &= 0.5 * ||X - WH||_{loss}^2
&+ alpha\\_W * l1\\_ratio * n\\_features * ||vec(W)||_1
&+ alpha\\_H * l1\\_ratio * n\\_samples * ||vec(H)||_1
&+ 0.5 * alpha\\_W * (1 - l1\\_ratio) * n\\_features * ||W||_{Fro}^2
&+ 0.5 * alpha\\_H * (1 - l1\\_ratio) * n\\_samples * ||H||_{Fro}^2,
where :math:`||A||_{Fro}^2 = \\sum_{i,j} A_{ij}^2` (Frobenius norm) and
:math:`||vec(A)||_1 = \\sum_{i,j} abs(A_{ij})` (Elementwise L1 norm)
The generic norm :math:`||X - WH||_{loss}^2` may represent
the Frobenius norm or another supported beta-divergence loss.
The choice between options is controlled by the `beta_loss` parameter.
The regularization terms are scaled by `n_features` for `W` and by `n_samples` for
`H` to keep their impact balanced with respect to one another and to the data fit
term as independent as possible of the size `n_samples` of the training set.
The objective function is minimized with an alternating minimization of W
and H. If H is given and update_H=False, it solves for W only.
Note that the transformed data is named W and the components matrix is named H. In
the NMF literature, the naming convention is usually the opposite since the data
matrix X is transposed.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Constant matrix.
W : array-like of shape (n_samples, n_components), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `update_H=False`, it is initialised as an array of zeros, unless
`solver='mu'`, then it is filled with values calculated by
`np.sqrt(X.mean() / self._n_components)`.
If `None`, uses the initialisation method specified in `init`.
H : array-like of shape (n_components, n_features), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `update_H=False`, it is used as a constant, to solve for W only.
If `None`, uses the initialisation method specified in `init`.
n_components : int or {'auto'} or None, default='auto'
Number of components. If `None`, all features are kept.
If `n_components='auto'`, the number of components is automatically inferred
from `W` or `H` shapes.
.. versionchanged:: 1.4
Added `'auto'` value.
.. versionchanged:: 1.6
Default value changed from `None` to `'auto'`.
init : {'random', 'nndsvd', 'nndsvda', 'nndsvdar', 'custom'}, default=None
Method used to initialize the procedure.
Valid options:
- None: 'nndsvda' if n_components < n_features, otherwise 'random'.
- 'random': non-negative random matrices, scaled with:
`sqrt(X.mean() / n_components)`
- 'nndsvd': Nonnegative Double Singular Value Decomposition (NNDSVD)
initialization (better for sparseness)
- 'nndsvda': NNDSVD with zeros filled with the average of X
(better when sparsity is not desired)
- 'nndsvdar': NNDSVD with zeros filled with small random values
(generally faster, less accurate alternative to NNDSVDa
for when sparsity is not desired)
- 'custom': If `update_H=True`, use custom matrices W and H which must both
be provided. If `update_H=False`, then only custom matrix H is used.
.. versionchanged:: 0.23
The default value of `init` changed from 'random' to None in 0.23.
.. versionchanged:: 1.1
When `init=None` and n_components is less than n_samples and n_features
defaults to `nndsvda` instead of `nndsvd`.
update_H : bool, default=True
Set to True, both W and H will be estimated from initial guesses.
Set to False, only W will be estimated.
solver : {'cd', 'mu'}, default='cd'
Numerical solver to use:
- 'cd' is a Coordinate Descent solver that uses Fast Hierarchical
Alternating Least Squares (Fast HALS).
- 'mu' is a Multiplicative Update solver.
.. versionadded:: 0.17
Coordinate Descent solver.
.. versionadded:: 0.19
Multiplicative Update solver.
beta_loss : float or {'frobenius', 'kullback-leibler', \
'itakura-saito'}, default='frobenius'
Beta divergence to be minimized, measuring the distance between X
and the dot product WH. Note that values different from 'frobenius'
(or 2) and 'kullback-leibler' (or 1) lead to significantly slower
fits. Note that for beta_loss <= 0 (or 'itakura-saito'), the input
matrix X cannot contain zeros. Used only in 'mu' solver.
.. versionadded:: 0.19
tol : float, default=1e-4
Tolerance of the stopping condition.
max_iter : int, default=200
Maximum number of iterations before timing out.
alpha_W : float, default=0.0
Constant that multiplies the regularization terms of `W`. Set it to zero
(default) to have no regularization on `W`.
.. versionadded:: 1.0
alpha_H : float or "same", default="same"
Constant that multiplies the regularization terms of `H`. Set it to zero to
have no regularization on `H`. If "same" (default), it takes the same value as
`alpha_W`.
.. versionadded:: 1.0
l1_ratio : float, default=0.0
The regularization mixing parameter, with 0 <= l1_ratio <= 1.
For l1_ratio = 0 the penalty is an elementwise L2 penalty
(aka Frobenius Norm).
For l1_ratio = 1 it is an elementwise L1 penalty.
For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2.
random_state : int, RandomState instance or None, default=None
Used for NMF initialisation (when ``init`` == 'nndsvdar' or
'random'), and in Coordinate Descent. Pass an int for reproducible
results across multiple function calls.
See :term:`Glossary <random_state>`.
verbose : int, default=0
The verbosity level.
shuffle : bool, default=False
If true, randomize the order of coordinates in the CD solver.
Returns
-------
W : ndarray of shape (n_samples, n_components)
Solution to the non-negative least squares problem.
H : ndarray of shape (n_components, n_features)
Solution to the non-negative least squares problem.
n_iter : int
Actual number of iterations.
References
----------
.. [1] :doi:`"Fast local algorithms for large scale nonnegative matrix and tensor
factorizations" <10.1587/transfun.E92.A.708>`
Cichocki, Andrzej, and P. H. A. N. Anh-Huy. IEICE transactions on fundamentals
of electronics, communications and computer sciences 92.3: 708-721, 2009.
.. [2] :doi:`"Algorithms for nonnegative matrix factorization with the
beta-divergence" <10.1162/NECO_a_00168>`
Fevotte, C., & Idier, J. (2011). Neural Computation, 23(9).
Examples
--------
>>> import numpy as np
>>> X = np.array([[1,1], [2, 1], [3, 1.2], [4, 1], [5, 0.8], [6, 1]])
>>> from sklearn.decomposition import non_negative_factorization
>>> W, H, n_iter = non_negative_factorization(
... X, n_components=2, init='random', random_state=0)
"""
est = NMF(
n_components=n_components,
init=init,
solver=solver,
beta_loss=beta_loss,
tol=tol,
max_iter=max_iter,
random_state=random_state,
alpha_W=alpha_W,
alpha_H=alpha_H,
l1_ratio=l1_ratio,
verbose=verbose,
shuffle=shuffle,
)
est._validate_params()
X = check_array(X, accept_sparse=("csr", "csc"), dtype=[np.float64, np.float32])
with config_context(assume_finite=True):
W, H, n_iter = est._fit_transform(X, W=W, H=H, update_H=update_H)
return W, H, n_iter
class _BaseNMF(ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator, ABC):
"""Base class for NMF and MiniBatchNMF."""
# This prevents ``set_split_inverse_transform`` to be generated for the
# non-standard ``Xt`` arg on ``inverse_transform``.
# TODO(1.7): remove when Xt is removed in v1.7 for inverse_transform
__metadata_request__inverse_transform = {"Xt": metadata_routing.UNUSED}
_parameter_constraints: dict = {
"n_components": [
Interval(Integral, 1, None, closed="left"),
None,
StrOptions({"auto"}),
],
"init": [
StrOptions({"random", "nndsvd", "nndsvda", "nndsvdar", "custom"}),
None,
],
"beta_loss": [
StrOptions({"frobenius", "kullback-leibler", "itakura-saito"}),
Real,
],
"tol": [Interval(Real, 0, None, closed="left")],
"max_iter": [Interval(Integral, 1, None, closed="left")],
"random_state": ["random_state"],
"alpha_W": [Interval(Real, 0, None, closed="left")],
"alpha_H": [Interval(Real, 0, None, closed="left"), StrOptions({"same"})],
"l1_ratio": [Interval(Real, 0, 1, closed="both")],
"verbose": ["verbose"],
}
def __init__(
self,
n_components="auto",
*,
init=None,
beta_loss="frobenius",
tol=1e-4,
max_iter=200,
random_state=None,
alpha_W=0.0,
alpha_H="same",
l1_ratio=0.0,
verbose=0,
):
self.n_components = n_components
self.init = init
self.beta_loss = beta_loss
self.tol = tol
self.max_iter = max_iter
self.random_state = random_state
self.alpha_W = alpha_W
self.alpha_H = alpha_H
self.l1_ratio = l1_ratio
self.verbose = verbose
def _check_params(self, X):
# n_components
self._n_components = self.n_components
if self._n_components is None:
self._n_components = X.shape[1]
# beta_loss
self._beta_loss = _beta_loss_to_float(self.beta_loss)
def _check_w_h(self, X, W, H, update_H):
"""Check W and H, or initialize them."""
n_samples, n_features = X.shape
if self.init == "custom" and update_H:
_check_init(H, (self._n_components, n_features), "NMF (input H)")
_check_init(W, (n_samples, self._n_components), "NMF (input W)")
if self._n_components == "auto":
self._n_components = H.shape[0]
if H.dtype != X.dtype or W.dtype != X.dtype:
raise TypeError(
"H and W should have the same dtype as X. Got "
"H.dtype = {} and W.dtype = {}.".format(H.dtype, W.dtype)
)
elif not update_H:
if W is not None:
warnings.warn(
"When update_H=False, the provided initial W is not used.",
RuntimeWarning,
)
_check_init(H, (self._n_components, n_features), "NMF (input H)")
if self._n_components == "auto":
self._n_components = H.shape[0]
if H.dtype != X.dtype:
raise TypeError(
"H should have the same dtype as X. Got H.dtype = {}.".format(
H.dtype
)
)
# 'mu' solver should not be initialized by zeros
if self.solver == "mu":
avg = np.sqrt(X.mean() / self._n_components)
W = np.full((n_samples, self._n_components), avg, dtype=X.dtype)
else:
W = np.zeros((n_samples, self._n_components), dtype=X.dtype)
else:
if W is not None or H is not None:
warnings.warn(
(
"When init!='custom', provided W or H are ignored. Set "
" init='custom' to use them as initialization."
),
RuntimeWarning,
)
if self._n_components == "auto":
self._n_components = X.shape[1]
W, H = _initialize_nmf(
X, self._n_components, init=self.init, random_state=self.random_state
)
return W, H
def _compute_regularization(self, X):
"""Compute scaled regularization terms."""
n_samples, n_features = X.shape
alpha_W = self.alpha_W
alpha_H = self.alpha_W if self.alpha_H == "same" else self.alpha_H
l1_reg_W = n_features * alpha_W * self.l1_ratio
l1_reg_H = n_samples * alpha_H * self.l1_ratio
l2_reg_W = n_features * alpha_W * (1.0 - self.l1_ratio)
l2_reg_H = n_samples * alpha_H * (1.0 - self.l1_ratio)
return l1_reg_W, l1_reg_H, l2_reg_W, l2_reg_H
def fit(self, X, y=None, **params):
"""Learn a NMF model for the data X.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Training vector, where `n_samples` is the number of samples
and `n_features` is the number of features.
y : Ignored
Not used, present for API consistency by convention.
**params : kwargs
Parameters (keyword arguments) and values passed to
the fit_transform instance.
Returns
-------
self : object
Returns the instance itself.
"""
# param validation is done in fit_transform
self.fit_transform(X, **params)
return self
def inverse_transform(self, X=None, *, Xt=None):
"""Transform data back to its original space.
.. versionadded:: 0.18
Parameters
----------
X : {ndarray, sparse matrix} of shape (n_samples, n_components)
Transformed data matrix.
Xt : {ndarray, sparse matrix} of shape (n_samples, n_components)
Transformed data matrix.
.. deprecated:: 1.5
`Xt` was deprecated in 1.5 and will be removed in 1.7. Use `X` instead.
Returns
-------
X : ndarray of shape (n_samples, n_features)
Returns a data matrix of the original shape.
"""
X = _deprecate_Xt_in_inverse_transform(X, Xt)
check_is_fitted(self)
return X @ self.components_
@property
def _n_features_out(self):
"""Number of transformed output features."""
return self.components_.shape[0]
def __sklearn_tags__(self):
tags = super().__sklearn_tags__()
tags.input_tags.positive_only = True
tags.input_tags.sparse = True
tags.transformer_tags.preserves_dtype = ["float64", "float32"]
return tags
class NMF(_BaseNMF):
"""Non-Negative Matrix Factorization (NMF).
Find two non-negative matrices, i.e. matrices with all non-negative elements, (W, H)
whose product approximates the non-negative matrix X. This factorization can be used
for example for dimensionality reduction, source separation or topic extraction.
The objective function is:
.. math::
L(W, H) &= 0.5 * ||X - WH||_{loss}^2
&+ alpha\\_W * l1\\_ratio * n\\_features * ||vec(W)||_1
&+ alpha\\_H * l1\\_ratio * n\\_samples * ||vec(H)||_1
&+ 0.5 * alpha\\_W * (1 - l1\\_ratio) * n\\_features * ||W||_{Fro}^2
&+ 0.5 * alpha\\_H * (1 - l1\\_ratio) * n\\_samples * ||H||_{Fro}^2,
where :math:`||A||_{Fro}^2 = \\sum_{i,j} A_{ij}^2` (Frobenius norm) and
:math:`||vec(A)||_1 = \\sum_{i,j} abs(A_{ij})` (Elementwise L1 norm).
The generic norm :math:`||X - WH||_{loss}` may represent
the Frobenius norm or another supported beta-divergence loss.
The choice between options is controlled by the `beta_loss` parameter.
The regularization terms are scaled by `n_features` for `W` and by `n_samples` for
`H` to keep their impact balanced with respect to one another and to the data fit
term as independent as possible of the size `n_samples` of the training set.
The objective function is minimized with an alternating minimization of W
and H.
Note that the transformed data is named W and the components matrix is named H. In
the NMF literature, the naming convention is usually the opposite since the data
matrix X is transposed.
Read more in the :ref:`User Guide <NMF>`.
Parameters
----------
n_components : int or {'auto'} or None, default='auto'
Number of components. If `None`, all features are kept.
If `n_components='auto'`, the number of components is automatically inferred
from W or H shapes.
.. versionchanged:: 1.4
Added `'auto'` value.
.. versionchanged:: 1.6
Default value changed from `None` to `'auto'`.
init : {'random', 'nndsvd', 'nndsvda', 'nndsvdar', 'custom'}, default=None
Method used to initialize the procedure.
Valid options:
- `None`: 'nndsvda' if n_components <= min(n_samples, n_features),
otherwise random.
- `'random'`: non-negative random matrices, scaled with:
`sqrt(X.mean() / n_components)`
- `'nndsvd'`: Nonnegative Double Singular Value Decomposition (NNDSVD)
initialization (better for sparseness)
- `'nndsvda'`: NNDSVD with zeros filled with the average of X
(better when sparsity is not desired)
- `'nndsvdar'` NNDSVD with zeros filled with small random values
(generally faster, less accurate alternative to NNDSVDa
for when sparsity is not desired)
- `'custom'`: Use custom matrices `W` and `H` which must both be provided.
.. versionchanged:: 1.1
When `init=None` and n_components is less than n_samples and n_features
defaults to `nndsvda` instead of `nndsvd`.
solver : {'cd', 'mu'}, default='cd'
Numerical solver to use:
- 'cd' is a Coordinate Descent solver.
- 'mu' is a Multiplicative Update solver.
.. versionadded:: 0.17
Coordinate Descent solver.
.. versionadded:: 0.19
Multiplicative Update solver.
beta_loss : float or {'frobenius', 'kullback-leibler', \
'itakura-saito'}, default='frobenius'
Beta divergence to be minimized, measuring the distance between X
and the dot product WH. Note that values different from 'frobenius'
(or 2) and 'kullback-leibler' (or 1) lead to significantly slower
fits. Note that for beta_loss <= 0 (or 'itakura-saito'), the input
matrix X cannot contain zeros. Used only in 'mu' solver.
.. versionadded:: 0.19
tol : float, default=1e-4
Tolerance of the stopping condition.
max_iter : int, default=200
Maximum number of iterations before timing out.
random_state : int, RandomState instance or None, default=None
Used for initialisation (when ``init`` == 'nndsvdar' or
'random'), and in Coordinate Descent. Pass an int for reproducible
results across multiple function calls.
See :term:`Glossary <random_state>`.
alpha_W : float, default=0.0
Constant that multiplies the regularization terms of `W`. Set it to zero
(default) to have no regularization on `W`.
.. versionadded:: 1.0
alpha_H : float or "same", default="same"
Constant that multiplies the regularization terms of `H`. Set it to zero to
have no regularization on `H`. If "same" (default), it takes the same value as
`alpha_W`.
.. versionadded:: 1.0
l1_ratio : float, default=0.0
The regularization mixing parameter, with 0 <= l1_ratio <= 1.
For l1_ratio = 0 the penalty is an elementwise L2 penalty
(aka Frobenius Norm).
For l1_ratio = 1 it is an elementwise L1 penalty.
For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2.
.. versionadded:: 0.17
Regularization parameter *l1_ratio* used in the Coordinate Descent
solver.
verbose : int, default=0
Whether to be verbose.
shuffle : bool, default=False
If true, randomize the order of coordinates in the CD solver.
.. versionadded:: 0.17
*shuffle* parameter used in the Coordinate Descent solver.
Attributes
----------
components_ : ndarray of shape (n_components, n_features)
Factorization matrix, sometimes called 'dictionary'.
n_components_ : int
The number of components. It is same as the `n_components` parameter
if it was given. Otherwise, it will be same as the number of
features.
reconstruction_err_ : float
Frobenius norm of the matrix difference, or beta-divergence, between
the training data ``X`` and the reconstructed data ``WH`` from
the fitted model.
n_iter_ : int
Actual number of iterations.
n_features_in_ : int
Number of features seen during :term:`fit`.
.. versionadded:: 0.24
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
See Also
--------
DictionaryLearning : Find a dictionary that sparsely encodes data.
MiniBatchSparsePCA : Mini-batch Sparse Principal Components Analysis.
PCA : Principal component analysis.
SparseCoder : Find a sparse representation of data from a fixed,
precomputed dictionary.
SparsePCA : Sparse Principal Components Analysis.
TruncatedSVD : Dimensionality reduction using truncated SVD.
References
----------
.. [1] :doi:`"Fast local algorithms for large scale nonnegative matrix and tensor
factorizations" <10.1587/transfun.E92.A.708>`
Cichocki, Andrzej, and P. H. A. N. Anh-Huy. IEICE transactions on fundamentals
of electronics, communications and computer sciences 92.3: 708-721, 2009.
.. [2] :doi:`"Algorithms for nonnegative matrix factorization with the
beta-divergence" <10.1162/NECO_a_00168>`
Fevotte, C., & Idier, J. (2011). Neural Computation, 23(9).
Examples
--------
>>> import numpy as np
>>> X = np.array([[1, 1], [2, 1], [3, 1.2], [4, 1], [5, 0.8], [6, 1]])
>>> from sklearn.decomposition import NMF
>>> model = NMF(n_components=2, init='random', random_state=0)
>>> W = model.fit_transform(X)
>>> H = model.components_
"""
_parameter_constraints: dict = {
**_BaseNMF._parameter_constraints,
"solver": [StrOptions({"mu", "cd"})],
"shuffle": ["boolean"],
}
def __init__(
self,
n_components="auto",
*,
init=None,
solver="cd",
beta_loss="frobenius",
tol=1e-4,
max_iter=200,
random_state=None,
alpha_W=0.0,
alpha_H="same",
l1_ratio=0.0,
verbose=0,
shuffle=False,
):
super().__init__(
n_components=n_components,
init=init,
beta_loss=beta_loss,
tol=tol,
max_iter=max_iter,
random_state=random_state,
alpha_W=alpha_W,
alpha_H=alpha_H,
l1_ratio=l1_ratio,
verbose=verbose,
)
self.solver = solver
self.shuffle = shuffle
def _check_params(self, X):
super()._check_params(X)
# solver
if self.solver != "mu" and self.beta_loss not in (2, "frobenius"):
# 'mu' is the only solver that handles other beta losses than 'frobenius'
raise ValueError(
f"Invalid beta_loss parameter: solver {self.solver!r} does not handle "
f"beta_loss = {self.beta_loss!r}"
)
if self.solver == "mu" and self.init == "nndsvd":
warnings.warn(
(
"The multiplicative update ('mu') solver cannot update "
"zeros present in the initialization, and so leads to "
"poorer results when used jointly with init='nndsvd'. "
"You may try init='nndsvda' or init='nndsvdar' instead."
),
UserWarning,
)
return self
@_fit_context(prefer_skip_nested_validation=True)
def fit_transform(self, X, y=None, W=None, H=None):
"""Learn a NMF model for the data X and returns the transformed data.
This is more efficient than calling fit followed by transform.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Training vector, where `n_samples` is the number of samples
and `n_features` is the number of features.
y : Ignored
Not used, present for API consistency by convention.
W : array-like of shape (n_samples, n_components), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `None`, uses the initialisation method specified in `init`.
H : array-like of shape (n_components, n_features), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `None`, uses the initialisation method specified in `init`.
Returns
-------
W : ndarray of shape (n_samples, n_components)
Transformed data.
"""
X = validate_data(
self, X, accept_sparse=("csr", "csc"), dtype=[np.float64, np.float32]
)
with config_context(assume_finite=True):
W, H, n_iter = self._fit_transform(X, W=W, H=H)
self.reconstruction_err_ = _beta_divergence(
X, W, H, self._beta_loss, square_root=True
)
self.n_components_ = H.shape[0]
self.components_ = H
self.n_iter_ = n_iter
return W
def _fit_transform(self, X, y=None, W=None, H=None, update_H=True):
"""Learn a NMF model for the data X and returns the transformed data.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Data matrix to be decomposed
y : Ignored
W : array-like of shape (n_samples, n_components), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `update_H=False`, it is initialised as an array of zeros, unless
`solver='mu'`, then it is filled with values calculated by
`np.sqrt(X.mean() / self._n_components)`.
If `None`, uses the initialisation method specified in `init`.
H : array-like of shape (n_components, n_features), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `update_H=False`, it is used as a constant, to solve for W only.
If `None`, uses the initialisation method specified in `init`.
update_H : bool, default=True
If True, both W and H will be estimated from initial guesses,
this corresponds to a call to the 'fit_transform' method.
If False, only W will be estimated, this corresponds to a call
to the 'transform' method.
Returns
-------
W : ndarray of shape (n_samples, n_components)
Transformed data.
H : ndarray of shape (n_components, n_features)
Factorization matrix, sometimes called 'dictionary'.
n_iter_ : int
Actual number of iterations.
"""
# check parameters
self._check_params(X)
if X.min() == 0 and self._beta_loss <= 0:
raise ValueError(
"When beta_loss <= 0 and X contains zeros, "
"the solver may diverge. Please add small values "
"to X, or use a positive beta_loss."
)
# initialize or check W and H
W, H = self._check_w_h(X, W, H, update_H)
# scale the regularization terms
l1_reg_W, l1_reg_H, l2_reg_W, l2_reg_H = self._compute_regularization(X)
if self.solver == "cd":
W, H, n_iter = _fit_coordinate_descent(
X,
W,
H,
self.tol,
self.max_iter,
l1_reg_W,
l1_reg_H,
l2_reg_W,
l2_reg_H,
update_H=update_H,
verbose=self.verbose,
shuffle=self.shuffle,
random_state=self.random_state,
)
elif self.solver == "mu":
W, H, n_iter, *_ = _fit_multiplicative_update(
X,
W,
H,
self._beta_loss,
self.max_iter,
self.tol,
l1_reg_W,
l1_reg_H,
l2_reg_W,
l2_reg_H,
update_H,
self.verbose,
)
else:
raise ValueError("Invalid solver parameter '%s'." % self.solver)
if n_iter == self.max_iter and self.tol > 0:
warnings.warn(
"Maximum number of iterations %d reached. Increase "
"it to improve convergence." % self.max_iter,
ConvergenceWarning,
)
return W, H, n_iter
def transform(self, X):
"""Transform the data X according to the fitted NMF model.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Training vector, where `n_samples` is the number of samples
and `n_features` is the number of features.
Returns
-------
W : ndarray of shape (n_samples, n_components)
Transformed data.
"""
check_is_fitted(self)
X = validate_data(
self,
X,
accept_sparse=("csr", "csc"),
dtype=[np.float64, np.float32],
reset=False,
ensure_non_negative=True,
)
with config_context(assume_finite=True):
W, *_ = self._fit_transform(X, H=self.components_, update_H=False)
return W
class MiniBatchNMF(_BaseNMF):
"""Mini-Batch Non-Negative Matrix Factorization (NMF).
.. versionadded:: 1.1
Find two non-negative matrices, i.e. matrices with all non-negative elements,
(`W`, `H`) whose product approximates the non-negative matrix `X`. This
factorization can be used for example for dimensionality reduction, source
separation or topic extraction.
The objective function is:
.. math::
L(W, H) &= 0.5 * ||X - WH||_{loss}^2
&+ alpha\\_W * l1\\_ratio * n\\_features * ||vec(W)||_1
&+ alpha\\_H * l1\\_ratio * n\\_samples * ||vec(H)||_1
&+ 0.5 * alpha\\_W * (1 - l1\\_ratio) * n\\_features * ||W||_{Fro}^2
&+ 0.5 * alpha\\_H * (1 - l1\\_ratio) * n\\_samples * ||H||_{Fro}^2,
where :math:`||A||_{Fro}^2 = \\sum_{i,j} A_{ij}^2` (Frobenius norm) and
:math:`||vec(A)||_1 = \\sum_{i,j} abs(A_{ij})` (Elementwise L1 norm).
The generic norm :math:`||X - WH||_{loss}^2` may represent
the Frobenius norm or another supported beta-divergence loss.
The choice between options is controlled by the `beta_loss` parameter.
The objective function is minimized with an alternating minimization of `W`
and `H`.
Note that the transformed data is named `W` and the components matrix is
named `H`. In the NMF literature, the naming convention is usually the opposite
since the data matrix `X` is transposed.
Read more in the :ref:`User Guide <MiniBatchNMF>`.
Parameters
----------
n_components : int or {'auto'} or None, default='auto'
Number of components. If `None`, all features are kept.
If `n_components='auto'`, the number of components is automatically inferred
from W or H shapes.
.. versionchanged:: 1.4
Added `'auto'` value.
.. versionchanged:: 1.6
Default value changed from `None` to `'auto'`.
init : {'random', 'nndsvd', 'nndsvda', 'nndsvdar', 'custom'}, default=None
Method used to initialize the procedure.
Valid options:
- `None`: 'nndsvda' if `n_components <= min(n_samples, n_features)`,
otherwise random.
- `'random'`: non-negative random matrices, scaled with:
`sqrt(X.mean() / n_components)`
- `'nndsvd'`: Nonnegative Double Singular Value Decomposition (NNDSVD)
initialization (better for sparseness).
- `'nndsvda'`: NNDSVD with zeros filled with the average of X
(better when sparsity is not desired).
- `'nndsvdar'` NNDSVD with zeros filled with small random values
(generally faster, less accurate alternative to NNDSVDa
for when sparsity is not desired).
- `'custom'`: Use custom matrices `W` and `H` which must both be provided.
batch_size : int, default=1024
Number of samples in each mini-batch. Large batch sizes
give better long-term convergence at the cost of a slower start.
beta_loss : float or {'frobenius', 'kullback-leibler', \
'itakura-saito'}, default='frobenius'
Beta divergence to be minimized, measuring the distance between `X`
and the dot product `WH`. Note that values different from 'frobenius'
(or 2) and 'kullback-leibler' (or 1) lead to significantly slower
fits. Note that for `beta_loss <= 0` (or 'itakura-saito'), the input
matrix `X` cannot contain zeros.
tol : float, default=1e-4
Control early stopping based on the norm of the differences in `H`
between 2 steps. To disable early stopping based on changes in `H`, set
`tol` to 0.0.
max_no_improvement : int, default=10
Control early stopping based on the consecutive number of mini batches
that does not yield an improvement on the smoothed cost function.
To disable convergence detection based on cost function, set
`max_no_improvement` to None.
max_iter : int, default=200
Maximum number of iterations over the complete dataset before
timing out.
alpha_W : float, default=0.0
Constant that multiplies the regularization terms of `W`. Set it to zero
(default) to have no regularization on `W`.
alpha_H : float or "same", default="same"
Constant that multiplies the regularization terms of `H`. Set it to zero to
have no regularization on `H`. If "same" (default), it takes the same value as
`alpha_W`.
l1_ratio : float, default=0.0
The regularization mixing parameter, with 0 <= l1_ratio <= 1.
For l1_ratio = 0 the penalty is an elementwise L2 penalty
(aka Frobenius Norm).
For l1_ratio = 1 it is an elementwise L1 penalty.
For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2.
forget_factor : float, default=0.7
Amount of rescaling of past information. Its value could be 1 with
finite datasets. Choosing values < 1 is recommended with online
learning as more recent batches will weight more than past batches.
fresh_restarts : bool, default=False
Whether to completely solve for W at each step. Doing fresh restarts will likely
lead to a better solution for a same number of iterations but it is much slower.
fresh_restarts_max_iter : int, default=30
Maximum number of iterations when solving for W at each step. Only used when
doing fresh restarts. These iterations may be stopped early based on a small
change of W controlled by `tol`.
transform_max_iter : int, default=None
Maximum number of iterations when solving for W at transform time.
If None, it defaults to `max_iter`.
random_state : int, RandomState instance or None, default=None
Used for initialisation (when ``init`` == 'nndsvdar' or
'random'), and in Coordinate Descent. Pass an int for reproducible
results across multiple function calls.
See :term:`Glossary <random_state>`.
verbose : bool, default=False
Whether to be verbose.
Attributes
----------
components_ : ndarray of shape (n_components, n_features)
Factorization matrix, sometimes called 'dictionary'.
n_components_ : int
The number of components. It is same as the `n_components` parameter
if it was given. Otherwise, it will be same as the number of
features.
reconstruction_err_ : float
Frobenius norm of the matrix difference, or beta-divergence, between
the training data `X` and the reconstructed data `WH` from
the fitted model.
n_iter_ : int
Actual number of started iterations over the whole dataset.
n_steps_ : int
Number of mini-batches processed.
n_features_in_ : int
Number of features seen during :term:`fit`.
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
See Also
--------
NMF : Non-negative matrix factorization.
MiniBatchDictionaryLearning : Finds a dictionary that can best be used to represent
data using a sparse code.
References
----------
.. [1] :doi:`"Fast local algorithms for large scale nonnegative matrix and tensor
factorizations" <10.1587/transfun.E92.A.708>`
Cichocki, Andrzej, and P. H. A. N. Anh-Huy. IEICE transactions on fundamentals
of electronics, communications and computer sciences 92.3: 708-721, 2009.
.. [2] :doi:`"Algorithms for nonnegative matrix factorization with the
beta-divergence" <10.1162/NECO_a_00168>`
Fevotte, C., & Idier, J. (2011). Neural Computation, 23(9).
.. [3] :doi:`"Online algorithms for nonnegative matrix factorization with the
Itakura-Saito divergence" <10.1109/ASPAA.2011.6082314>`
Lefevre, A., Bach, F., Fevotte, C. (2011). WASPA.
Examples
--------
>>> import numpy as np
>>> X = np.array([[1, 1], [2, 1], [3, 1.2], [4, 1], [5, 0.8], [6, 1]])
>>> from sklearn.decomposition import MiniBatchNMF
>>> model = MiniBatchNMF(n_components=2, init='random', random_state=0)
>>> W = model.fit_transform(X)
>>> H = model.components_
"""
_parameter_constraints: dict = {
**_BaseNMF._parameter_constraints,
"max_no_improvement": [Interval(Integral, 1, None, closed="left"), None],
"batch_size": [Interval(Integral, 1, None, closed="left")],
"forget_factor": [Interval(Real, 0, 1, closed="both")],
"fresh_restarts": ["boolean"],
"fresh_restarts_max_iter": [Interval(Integral, 1, None, closed="left")],
"transform_max_iter": [Interval(Integral, 1, None, closed="left"), None],
}
def __init__(
self,
n_components="auto",
*,
init=None,
batch_size=1024,
beta_loss="frobenius",
tol=1e-4,
max_no_improvement=10,
max_iter=200,
alpha_W=0.0,
alpha_H="same",
l1_ratio=0.0,
forget_factor=0.7,
fresh_restarts=False,
fresh_restarts_max_iter=30,
transform_max_iter=None,
random_state=None,
verbose=0,
):
super().__init__(
n_components=n_components,
init=init,
beta_loss=beta_loss,
tol=tol,
max_iter=max_iter,
random_state=random_state,
alpha_W=alpha_W,
alpha_H=alpha_H,
l1_ratio=l1_ratio,
verbose=verbose,
)
self.max_no_improvement = max_no_improvement
self.batch_size = batch_size
self.forget_factor = forget_factor
self.fresh_restarts = fresh_restarts
self.fresh_restarts_max_iter = fresh_restarts_max_iter
self.transform_max_iter = transform_max_iter
def _check_params(self, X):
super()._check_params(X)
# batch_size
self._batch_size = min(self.batch_size, X.shape[0])
# forget_factor
self._rho = self.forget_factor ** (self._batch_size / X.shape[0])
# gamma for Maximization-Minimization (MM) algorithm [Fevotte 2011]
if self._beta_loss < 1:
self._gamma = 1.0 / (2.0 - self._beta_loss)
elif self._beta_loss > 2:
self._gamma = 1.0 / (self._beta_loss - 1.0)
else:
self._gamma = 1.0
# transform_max_iter
self._transform_max_iter = (
self.max_iter
if self.transform_max_iter is None
else self.transform_max_iter
)
return self
def _solve_W(self, X, H, max_iter):
"""Minimize the objective function w.r.t W.
Update W with H being fixed, until convergence. This is the heart
of `transform` but it's also used during `fit` when doing fresh restarts.
"""
avg = np.sqrt(X.mean() / self._n_components)
W = np.full((X.shape[0], self._n_components), avg, dtype=X.dtype)
W_buffer = W.copy()
# Get scaled regularization terms. Done for each minibatch to take into account
# variable sizes of minibatches.
l1_reg_W, _, l2_reg_W, _ = self._compute_regularization(X)
for _ in range(max_iter):
W, *_ = _multiplicative_update_w(
X, W, H, self._beta_loss, l1_reg_W, l2_reg_W, self._gamma
)
W_diff = linalg.norm(W - W_buffer) / linalg.norm(W)
if self.tol > 0 and W_diff <= self.tol:
break
W_buffer[:] = W
return W
def _minibatch_step(self, X, W, H, update_H):
"""Perform the update of W and H for one minibatch."""
batch_size = X.shape[0]
# get scaled regularization terms. Done for each minibatch to take into account
# variable sizes of minibatches.
l1_reg_W, l1_reg_H, l2_reg_W, l2_reg_H = self._compute_regularization(X)
# update W
if self.fresh_restarts or W is None:
W = self._solve_W(X, H, self.fresh_restarts_max_iter)
else:
W, *_ = _multiplicative_update_w(
X, W, H, self._beta_loss, l1_reg_W, l2_reg_W, self._gamma
)
# necessary for stability with beta_loss < 1
if self._beta_loss < 1:
W[W < np.finfo(np.float64).eps] = 0.0
batch_cost = (
_beta_divergence(X, W, H, self._beta_loss)
+ l1_reg_W * W.sum()
+ l1_reg_H * H.sum()
+ l2_reg_W * (W**2).sum()
+ l2_reg_H * (H**2).sum()
) / batch_size
# update H (only at fit or fit_transform)
if update_H:
H[:] = _multiplicative_update_h(
X,
W,
H,
beta_loss=self._beta_loss,
l1_reg_H=l1_reg_H,
l2_reg_H=l2_reg_H,
gamma=self._gamma,
A=self._components_numerator,
B=self._components_denominator,
rho=self._rho,
)
# necessary for stability with beta_loss < 1
if self._beta_loss <= 1:
H[H < np.finfo(np.float64).eps] = 0.0
return batch_cost
def _minibatch_convergence(
self, X, batch_cost, H, H_buffer, n_samples, step, n_steps
):
"""Helper function to encapsulate the early stopping logic"""
batch_size = X.shape[0]
# counts steps starting from 1 for user friendly verbose mode.
step = step + 1
# Ignore first iteration because H is not updated yet.
if step == 1:
if self.verbose:
print(f"Minibatch step {step}/{n_steps}: mean batch cost: {batch_cost}")
return False
# Compute an Exponentially Weighted Average of the cost function to
# monitor the convergence while discarding minibatch-local stochastic
# variability: https://en.wikipedia.org/wiki/Moving_average
if self._ewa_cost is None:
self._ewa_cost = batch_cost
else:
alpha = batch_size / (n_samples + 1)
alpha = min(alpha, 1)
self._ewa_cost = self._ewa_cost * (1 - alpha) + batch_cost * alpha
# Log progress to be able to monitor convergence
if self.verbose:
print(
f"Minibatch step {step}/{n_steps}: mean batch cost: "
f"{batch_cost}, ewa cost: {self._ewa_cost}"
)
# Early stopping based on change of H
H_diff = linalg.norm(H - H_buffer) / linalg.norm(H)
if self.tol > 0 and H_diff <= self.tol:
if self.verbose:
print(f"Converged (small H change) at step {step}/{n_steps}")
return True
# Early stopping heuristic due to lack of improvement on smoothed
# cost function
if self._ewa_cost_min is None or self._ewa_cost < self._ewa_cost_min:
self._no_improvement = 0
self._ewa_cost_min = self._ewa_cost
else:
self._no_improvement += 1
if (
self.max_no_improvement is not None
and self._no_improvement >= self.max_no_improvement
):
if self.verbose:
print(
"Converged (lack of improvement in objective function) "
f"at step {step}/{n_steps}"
)
return True
return False
@_fit_context(prefer_skip_nested_validation=True)
def fit_transform(self, X, y=None, W=None, H=None):
"""Learn a NMF model for the data X and returns the transformed data.
This is more efficient than calling fit followed by transform.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Data matrix to be decomposed.
y : Ignored
Not used, present here for API consistency by convention.
W : array-like of shape (n_samples, n_components), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `None`, uses the initialisation method specified in `init`.
H : array-like of shape (n_components, n_features), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `None`, uses the initialisation method specified in `init`.
Returns
-------
W : ndarray of shape (n_samples, n_components)
Transformed data.
"""
X = validate_data(
self, X, accept_sparse=("csr", "csc"), dtype=[np.float64, np.float32]
)
with config_context(assume_finite=True):
W, H, n_iter, n_steps = self._fit_transform(X, W=W, H=H)
self.reconstruction_err_ = _beta_divergence(
X, W, H, self._beta_loss, square_root=True
)
self.n_components_ = H.shape[0]
self.components_ = H
self.n_iter_ = n_iter
self.n_steps_ = n_steps
return W
def _fit_transform(self, X, W=None, H=None, update_H=True):
"""Learn a NMF model for the data X and returns the transformed data.
Parameters
----------
X : {ndarray, sparse matrix} of shape (n_samples, n_features)
Data matrix to be decomposed.
W : array-like of shape (n_samples, n_components), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `update_H=False`, it is initialised as an array of zeros, unless
`solver='mu'`, then it is filled with values calculated by
`np.sqrt(X.mean() / self._n_components)`.
If `None`, uses the initialisation method specified in `init`.
H : array-like of shape (n_components, n_features), default=None
If `init='custom'`, it is used as initial guess for the solution.
If `update_H=False`, it is used as a constant, to solve for W only.
If `None`, uses the initialisation method specified in `init`.
update_H : bool, default=True
If True, both W and H will be estimated from initial guesses,
this corresponds to a call to the `fit_transform` method.
If False, only W will be estimated, this corresponds to a call
to the `transform` method.
Returns
-------
W : ndarray of shape (n_samples, n_components)
Transformed data.
H : ndarray of shape (n_components, n_features)
Factorization matrix, sometimes called 'dictionary'.
n_iter : int
Actual number of started iterations over the whole dataset.
n_steps : int
Number of mini-batches processed.
"""
check_non_negative(X, "MiniBatchNMF (input X)")
self._check_params(X)
if X.min() == 0 and self._beta_loss <= 0:
raise ValueError(
"When beta_loss <= 0 and X contains zeros, "
"the solver may diverge. Please add small values "
"to X, or use a positive beta_loss."
)
n_samples = X.shape[0]
# initialize or check W and H
W, H = self._check_w_h(X, W, H, update_H)
H_buffer = H.copy()
# Initialize auxiliary matrices
self._components_numerator = H.copy()
self._components_denominator = np.ones(H.shape, dtype=H.dtype)
# Attributes to monitor the convergence
self._ewa_cost = None
self._ewa_cost_min = None
self._no_improvement = 0
batches = gen_batches(n_samples, self._batch_size)
batches = itertools.cycle(batches)
n_steps_per_iter = int(np.ceil(n_samples / self._batch_size))
n_steps = self.max_iter * n_steps_per_iter
for i, batch in zip(range(n_steps), batches):
batch_cost = self._minibatch_step(X[batch], W[batch], H, update_H)
if update_H and self._minibatch_convergence(
X[batch], batch_cost, H, H_buffer, n_samples, i, n_steps
):
break
H_buffer[:] = H
if self.fresh_restarts:
W = self._solve_W(X, H, self._transform_max_iter)
n_steps = i + 1
n_iter = int(np.ceil(n_steps / n_steps_per_iter))
if n_iter == self.max_iter and self.tol > 0:
warnings.warn(
(
f"Maximum number of iterations {self.max_iter} reached. "
"Increase it to improve convergence."
),
ConvergenceWarning,
)
return W, H, n_iter, n_steps
def transform(self, X):
"""Transform the data X according to the fitted MiniBatchNMF model.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Data matrix to be transformed by the model.
Returns
-------
W : ndarray of shape (n_samples, n_components)
Transformed data.
"""
check_is_fitted(self)
X = validate_data(
self,
X,
accept_sparse=("csr", "csc"),
dtype=[np.float64, np.float32],
reset=False,
)
W = self._solve_W(X, self.components_, self._transform_max_iter)
return W
@_fit_context(prefer_skip_nested_validation=True)
def partial_fit(self, X, y=None, W=None, H=None):
"""Update the model using the data in `X` as a mini-batch.
This method is expected to be called several times consecutively
on different chunks of a dataset so as to implement out-of-core
or online learning.
This is especially useful when the whole dataset is too big to fit in
memory at once (see :ref:`scaling_strategies`).
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Data matrix to be decomposed.
y : Ignored
Not used, present here for API consistency by convention.
W : array-like of shape (n_samples, n_components), default=None
If `init='custom'`, it is used as initial guess for the solution.
Only used for the first call to `partial_fit`.
H : array-like of shape (n_components, n_features), default=None
If `init='custom'`, it is used as initial guess for the solution.
Only used for the first call to `partial_fit`.
Returns
-------
self
Returns the instance itself.
"""
has_components = hasattr(self, "components_")
X = validate_data(
self,
X,
accept_sparse=("csr", "csc"),
dtype=[np.float64, np.float32],
reset=not has_components,
)
if not has_components:
# This instance has not been fitted yet (fit or partial_fit)
self._check_params(X)
_, H = self._check_w_h(X, W=W, H=H, update_H=True)
self._components_numerator = H.copy()
self._components_denominator = np.ones(H.shape, dtype=H.dtype)
self.n_steps_ = 0
else:
H = self.components_
self._minibatch_step(X, None, H, update_H=True)
self.n_components_ = H.shape[0]
self.components_ = H
self.n_steps_ += 1
return self
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